
theorem Th30:
  for L be Abelian add-associative right_zeroed
  right_complementable left-distributive non empty doubleLoopStr for p,q,r be
  sequence of L holds (p+q)*'r = p*'r+q*'r
proof
  let L be Abelian add-associative right_zeroed right_complementable
  left-distributive non empty doubleLoopStr;
  let p,q,r be sequence of L;
  now
    let i be Element of NAT;
    consider r1 be FinSequence of the carrier of L such that
A1: len r1 = i+1 and
A2: ((p+q)*'r).i = Sum r1 and
A3: for k be Element of NAT st k in dom r1 holds r1.k = (p+q).(k-'1) *
    r.(i+1-'k) by Def9;
A4: dom r1 = Seg(i+1) by A1,FINSEQ_1:def 3;
    consider r3 be FinSequence of the carrier of L such that
A5: len r3 = i+1 and
A6: (q*'r).i = Sum r3 and
A7: for k be Element of NAT st k in dom r3 holds r3.k = q.(k-'1) * r.
    (i+1-'k) by Def9;
    consider r2 be FinSequence of the carrier of L such that
A8: len r2 = i+1 and
A9: (p*'r).i = Sum r2 and
A10: for k be Element of NAT st k in dom r2 holds r2.k = p.(k-'1) * r.(
    i+1-'k) by Def9;
    reconsider r29=r2, r39=r3 as Element of (i+1)-tuples_on (the carrier of L)
    by A8,A5,FINSEQ_2:92;
A11: len (r29 + r39) = i+1 by CARD_1:def 7;
    now
      let k be Nat;
      assume
A12:  k in dom r1;
      then
A13:  k in dom (r2 + r3) by A11,A4,FINSEQ_1:def 3;
      k in dom r3 by A5,A4,A12,FINSEQ_1:def 3;
      then
A14:  r3.k = q.(k-'1) * r.(i+1-'k) by A7;
      k in dom r2 by A8,A4,A12,FINSEQ_1:def 3;
      then
A15:  r2.k = p.(k-'1) * r.(i+1-'k) by A10;
      thus r1.k = (p+q).(k-'1) * r.(i+1-'k) by A3,A12
        .= (p.(k-'1) + q.(k-'1)) * r.(i+1-'k) by NORMSP_1:def 2
        .= p.(k-'1) * r.(i+1-'k) + q.(k-'1) * r.(i+1-'k) by VECTSP_1:def 3
        .= (r2 + r3).k by A13,A15,A14,FVSUM_1:17;
    end;
    then Sum r1 = Sum(r29 + r39) by A1,A11,FINSEQ_2:9
      .= Sum r2 + Sum r3 by FVSUM_1:76;
    hence ((p+q)*'r).i = (p*'r+q*'r).i by A2,A9,A6,NORMSP_1:def 2;
  end;
  hence thesis;
end;
